
Next, by the well-known entropy inequalities for subgaussian process [1]}, we have
\(\mathbb {E}_{\varepsilon }\big [\hat{R}_n(\mathcal {G}_N)\big ]&\le \inf _{g \in \mathcal {G}_n}\mathbb {E}_{\varepsilon } \bigg |n^{-1}\sum _{j=1}^n \varepsilon _j g(X_j)\bigg |\nonumber \\&\qquad + \frac{c}{\sqrt{n}}\int _0^{(2\sup _{g\in \mathcal {G}_n} P_n g^2)^{1/2}}H_{d_{P_n,2}}^{1/2}(\mathcal {G}_N;u)du \) 
